Despite centuries of observation and decades of theoretical work, the ~11 year solar magnetic sunspot cycle remains one of the longest-standing unsolved problems in astrophysics. Additional insights can be made using synoptic observations of proxies for magnetism in other stars, where the varied stellar properties set the conditions for separate "dynamo experiments." We examine the decadal-scale variability in Ca II H & K emission of the Sun and a set of 26 solar analog stars within ~5% of the solar effective temperature but with varied mean rotation. Using a quantitative metric for determining cycle quality, we find that cycles of the highest quality—like the Sun's—occur in the stars with slower rotation and lower mean activity. Reexamining the results of a larger set of ~100 stars from the Baliunas et al. 1995 study of Mount Wilson H & K emission, we find again that the highest quality cycles occur for low activity and high Rossby number, the ratio of the rotation period to the convective turnover time. Guided by these observations, we propose the hypothesis that Sun-like variability—either a clean, monoperiodic cycle or flat activity analogous to the Maunder Minimum—occurs in G- and K- type main-sequence stars if and only if the Noyes et al. 1984 semi-empirical Rossby number is larger than 1.5, or equivalently if the star is on the low-activity side of the Vaughan-Preston gap. These results demonstrate the critical role of the Rossby number in determining the behavior of stellar dynamos.